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Modeling of hyperspectral data with non-Gaussian distributions is gaining popularity in recent years. Such modeling mostly concentrates on attempts to describe a distribution, or its tails, of all image spectra. In this paper, we recognize that the presence of major materials in the image scene is likely to exhibit nonrandomness and only the remaining variability due to noise, or other factors, would exhibit random behavior. Hence, we assume a linear mixing model with a structured background, and we investigate various distributional models for the error term in that model. We propose one model based on the multivariate

The following linear mixing model (with a structured background) is often used in hyperspectral imaging literature [

In this paper, we want to address the question whether model (

The multivariate

In Section

In this paper, we are going to assume that the abundances of all materials sum up to 1, that is,

We also assume that no a priori information is available about the spectral signatures in

Let us denote by

In order to build a model of the form (

Numerical results in Sections

Color rendering of the cluttered AVIRIS urban scene in Rochester, NY, USA, used in this paper.

Here we investigate two families of distributions as models for the distributions of

Select as

Select all dimensions above the threshold for the deterministic part of the model (possibly including dimension 67) and use the remaining dimensions for the error term. From the point of view of notation in Section

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Figure

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The second potential family of distributions for modeling the distributions of

Since

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Figure

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The model fitting discussed in the previous section accounted only for the marginal distributions of

Base 10 log values of the chi-square statistic (for testing the multivariate fit to

Figure

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The second multivariate model that we want to discuss is based on the standard

The variance-covariance matrix of this distribution is equal to

This distribution is spherically contoured in the sphered coordinates and elliptically contoured in the original coordinates. All marginal distributions of the scaled multivariate

If

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In the two previous sections, we used an AVIRIS image as an example to demonstrate how to fit a linear mixing model with the

Color rendering of the 280 by 800 pixel HyMap Cooke City image (see [

In order to investigate the marginal distributions, we now follow the ideas and notation of Section

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Figure

Base 10 log values of the chi-square statistic (for testing the fit to the exponential power distribution) plotted versus the shape parameter

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Base 10 log values of the chi-square statistic (for testing the fit to the scaled

Figure

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So far, we discussed only the marginal distributions of

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Regarding the final conclusion about the model fit, we need to keep in mind that this is just one test of multivariate fit, and it needs to be considered together with the results shown in Figure

Since almost all chi-square values in Figure

The shape parameter

We have also checked the fit to the multivariate

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In this paper, we investigated various distributional models for the error term in the linear mixing model with a structured background. The models were tested on two datasets. One was an AVIRIS image and the other one was a subimage of a forest area in a HyMap Cooke City image. The first proposed model was based on independent components following an exponential power distribution. The model fitted reasonably well to both datasets in terms of modeling marginal distributions. For the AVIRIS image, the fit of the joint distribution worked quite well for a small number of components, but not for a larger number. For the forest area in the Cooke City image, the fit with the joint distribution of independent exponential power distributions worked very well. This successful modeling might be largely due to this dataset being a fairly homogenous set of spectra.

The second model was based on the multivariate